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The Dolbeault operator on Hermitian spin surfaces. (English) Zbl 0987.53011
Authors’ summary: We consider the Dolbeault operator \(\sqrt 2(\overline\partial+ \overline\partial^*)\) of \(K^{{1\over 2}}\) – the square root of the canonical line bundle which determines the spin structure of a compact Hermitian spin surface \((M,g,J)\). We prove that all cohomology groups \(H^i(M,{\mathcal O}(K^{{1\over 2}}))\) vanish if the scalar curvature of \(g\) is nonnegative and non-identically zero. As a consequence, zero is not an eigenvalue of the Dolbeault operator. Moreover, we estimate the first eigenvalue of the Dolbeault operator when the conformal scalar curvature \(k\) is nonnegative and when \(k\) is positive. In the first case we give a complete list of limiting manifolds and in the second we give non-Kähler examples of limiting manifolds.

MSC:
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B35 Local differential geometry of Hermitian and Kählerian structures
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